Everton's Marouane Fellaini steers in his side's fifth and final goal in their third-round FA Cup tie at Cheltenham. Photograph: Dan Rowley/Colorsport/Corbis

It was a night when Everton barely offered the briefest sense they might be ripe for an upset. David Moyes packed his team with just about all of the usual cast, resisting any temptation to make wholesale changes, and their performance could probably be summed up by the reaction of the Cheltenham supporters late-on, when the manager withdrew Leighton Baines and Leon Osman. On both occasions there was rich applause from the home stands in recognition of the pair's contribution to a straightforward victory.

Osman had scored the pick of the goals with some lovely footwork and anticipation early in the second half. Nikica Jelavic's seventh goal of the season, followed by a Baines penalty, had put Everton in control during the first half and by the end it was difficult to recall a single moment when they had given their opponents any real encouragement. Even when Russell Penn, Cheltenham's best player, invigorated their first sell-out crowd since 2006 by making it 3-1, Everton quickly put them back in their place.

Seamus Coleman clipped in a clever finish and the prize of a fourth-round tie at Sunderland or Bolton Wanderers was secure well before Marouane Fellaini completed the win in the final minutes.

Whaddon Road, as the locals still know it, provided a nice setting for a third-round tie under the floodlights, with its uneven pitch, the advertising hoardings for "Tony Mustoe's Carpets" and the row of trees that are visible behind the little terrace at one end of the pitch.

Yet Moyes had taken great care not to underestimate their opponents. Steven Pienaar was not among the travelling squad but otherwise it was a full-strength XI for the Premier League's fifth-placed side. Cheltenham, fourth in League Two, were quick to the ball and full of hard running but the gulf in talent was considerable.

"We did a really good job," Moyes said. "I wanted to play the strongest possible team. I was disappointed earlier in the season when we lost our League Cup tie against Leeds and some of the players I put in were not up to scratch. I wasn't going to make the same mistake again. It might have looked a bad decision if we'd picked up any injuries. But we're not involved in the Europa League or the League Cup so I think we should give it a go in the FA Cup and do the best we can."

They scored with their first meaningful attack, Jelavic following up after Fellaini's shot had come back off a post, and the prospect of a giantkilling was all but extinguished after 21 minutes when the referee, Kevin Friend, decided Alan Bennett had been climbing over Fellaini inside the penalty area, giving Baines the chance to score for a second successive match. In between, a curling free-kick from the left-back had brought a splendid save from the Cheltenham goalkeeper, Scott Brown, and Jelavic had fired a good chance wide from Fellaini's knockdown.

Everton had immediately set about getting the message across that they were not the kind of team lower-league opposition should consider vulnerable in such humble surroundings. But Billy Jones, the Cheltenham left-back, will have enjoyed the moment when he nutmegged Victor Anichebe in the first half and Penn deserved his goal, beating Tim Howard at the near post to continue Everton's biggest problem this season: keeping clean sheets.

Cheltenham, however, defended far too generously and Everton played with a sense of control as soon as they had taken the lead. There was never really a point when there was incessant pressure on the home goal, but there was also a strong suspicion the top-division side were playing within themselves.

Four minutes after the restart Anichebe's pass picked out Baines on the left and Cheltenham's right-back, Sido Jombati, was guilty of losing his man. Baines, such a terrific attacker in those positions, cut the ball into Osman's path and the midfielder had the presence of mind to delay his shot, slipping between two defenders with a change of direction and beating Brown with a left-foot shot.

It was a lovely goal, demonstrating the mix of skill and composure that has helped Osman break into the England team, and at that stage it might have been understandable if Cheltenham had gone into damage-limitation mode. Instead they surprised everyone, perhaps even themselves, by pulling one back within two minutes. Kaid Mohamed played the pass and Penn fired in a left-foot effort.

The problem for Cheltenham was that they looked susceptible just about every time their opponents attacked in numbers. Just before the hour Anichebe picked out Coleman running through from right-back and the defender clipped a shot over the oncoming Brown.

Anichebe, causing all sorts of problems, was also involved for the final goal, Fellaini putting it away on the half-volley to cap another impressive performance from the Belgian. "They showed a lot of respect for the FA Cup putting out that team," Mark Yates, the Cheltenham manager, reflected. "We can be proud of how we played but it's a big learning curve."